User:Eufalesio/Ultimate: Difference between revisions

is 41&53&217, with mapping [⟨1 0 0 25 -33 -13], ⟨0 1 0 -14 23 12], ⟨0 0 1 0 0 -1]]. It's otherwise known by in the wiki as ''[[cassaschismic]]'' (technical info inside), also [[User:Eufalesio/Important Tables#Temperament properties of Ultimate edos (I care about)|here]]; but I will simply call it '''Ultimate'''. My reasoning of this will become clear. Or at least, I expect you to understand why it's clear in my mind.

* [[Apotome]] = [[77/72]] (salozoma tempered out)

* [[77/72]] = [[27/26]] * [[36/35]] (eufalesma tempered out)

* [[Pythagorean comma|Poma]] = [[64/63]] (sasaruma tempered out)

* 2 pomas = [[33/32]] (salururuma tempered out)

* [[36/35]] = [[1053/1024]] (minisma tempered out)

* [[32/27]] = [[11/10]] * [[14/13]] (sinaisma tempered out)

The [[pergen]] is (P8, P5, ^1), where ^1 is the "minicomma"; a 3~5c interval that represents 385/384, 352/351, 5120/5103, 513/512, the layoma, etc. 4:5:6:7:9:11:13 is notated as P1 ^d4 P5 dd8 M9 AAA9 vAAA4. Using pomas (pythagorean commas) improves this notation. ''(to do: write out this chord with pomas)''

Here is a quick compressed cheat sheet of octave-reduced intervals. This is a MASSIVE simplification with many (infinitely many) intervals left out for the sake of brevity. For every entry here, ''(to do: complete this sentence)''

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! colspan="5" |minicomma-span

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|'''5/4'''

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The chain of fifths is a very important framework historically. It's been in Western music THE way to think about everything all the way from plainchant to Renaissance meantone temperaments to the modern day; where the 12-pitch-class circle of fifths is taught; 12edo, a massively over-represented tuning. It has a bit of a bad reputation in the xen circles, but the more I researched, the more I realized it is a '''paragon''', and that its position nowadays is very much well earned.  

41edo introduces the [[cassandra]] framework, which thanks to its incredibly accurate fifths, introduces the poma as an accidental. In cassandra, the poma acts as 81/80~64/63 at around ~29c, and two of them add to 1053/1024~33/32~36/35. 41edo is a bit overtempered but notable as a proportionally coarse but good system. 53edo has practically pure fifths and very good p5 (prime 5) and p13 (prime 13), but p7 and p11 are tuned worse.  

However, if you forgo p5 and p13 for the chain of fifths, you end up with [[gary]]. Gary is a serendipitous temperament, the same as cassandra but optimized for the [[zala]]. 135edo is an essentially perfect tuning,  reaching sub-cent levels of error in the subgroup, with a poma of 26.<u>6</u> c.

Ultimate is not just an extension of the concept, but what I believe to be the '''end''' of that extension. Ultimate adds an independent minicomma generator for p5 and p13, which acts as 385/384, ~352/351, ~5120/5103, layoma, ... etc, being around 4.4c. It doesn't begin to make sense up until 217edo, but 270edo and 311edo are arguably the best tunings. 217edo is alright, but not the best, which would make it a waste of time if it weren't a multiple of 31edo.

270edo and 311edo inherit a chain of fifths that is consistent with cassandra, which itself is an extension of the circle of fifths. The only addition is a single edostep, and respectively, the entire 13-limit is tuned to unfathomable precision, and the 41-limit is fully accessible and very well tuned. However, I prefer sticking to the 13-limit, so 270edo is an optimal equal tuning.

The coarsest tuning that makes sense. It can be written just with sharps and flats, since the poma and the minicomma are tempered out in all its possible expressions. 12e because patent val tunes 11/8 as a tritone, not fourth. The cassandra mapping is based on 11/8 as a kind of fourth, not tritone. Either way, p11 is NOT there. Consider it an extremely coarse [[yazatha]] tuning.

The coarsest cassandra tuning. It can be written with sharps and flats, plus ↑ and '''↓''' for the pomas. In the case of 41edo, there is no need for double pomas, because the apotome can be split in half. Thus, half sharps and half flats can be used instead of two pomas. ONLY in 41edo. Ideal for 11-limit pieces with acoustic instruments, like the well known [[Kite guitar]], albeit, it is not a cassandra layout, but [[Skip fretting system 41 2 13]]. The cassandra layout is [[skip fretting system 41 3 7]].

Another good cassandra tuning. Just like 41edo, It can be written with sharps and flats, plus ↑ and '''↓''' for the pomas. The poma can be doubled into ⇑ and '''⇓''' to reach p11 and p13. It is playable and around the extremum possible inside the [[Lumatone mapping for 103edo|Lumatone]], which despite having a p7 and p11 that are not too well tuned; it has good 13-limit capabilities. It can be used in a guitar with the [[skip freting system 53 4 9]].

Best cassandra tuning. Just like 53edo, it can be written with sharps and flats, plus ↑ and '''↓''' for the pomas. The poma can be doubled into ⇑ and '''⇓''' to reach p11 and p13. Since the chain takes much longer to close, / and \ may be used to raise or lower by a half-poma. (Note that a half-poma implies a half-octave, and thus an even-numbered edo.) This tuning is optimal and technically usable in the Lumatone, but only as a subset, requiring more than one preset to reach within the [[Standard Lumatone mapping for Pythagorean]]. The cassandra layout can be used in a guitar with the [[skip fretting system 94 7 16]]. However, in a 6-string guitar there will be no other unisons.

They all work much the same way. They can be written with sharps and flats, ↑ and '''↓''' for the pomas, ⇑ and '''⇓''' for doubled pomas, and the addition of ^ and v for the minicomma, taken directly from the [[Kite's ups and downs notation|ups-and-downs notation.]] This is completely unfeasible to use with a Lumatone, with any acoustic instrument, isomorphically; though, it can still be used in a DAW without much problem. Because ultimate is rank-3, the layout is 3D and thus it is impossible to play on a flat surface, requiring some sort of eldritch holographic "keyspace".

217edo could ''in theory'' be used with binary valve or key systems in woodwinds, granted they have the intonation precision to reliably hit pitches within a maximum error of 2.76 cents. Which I know won't happen. Best course would be to tune the instrument to 31edo plus a slide to nudge everything into the right place, but that's not Ultimate. That's [[birds]].

It is possible to forgo edos altogether and use Ultimate as is, providing the absolute best tuning possible. However, it's a rank-3 system. It has no extra unisons unlike the equal tunings. This is reserved for when 270edo is not just enough, and beating is something to avoid as much as possible.